Bone sharing weights with another bone?
Alright so I came to a solution but it's not exact. This is more of a, "Soft" fix. In other words it's more of a workaround rather than an actual solution. I explain my findings in complete detail in this video I made for it:

https://www.youtube.com/watch?v=wW3u6qJk2TQ

For those who cannot watch the video, basically both the Neck Bone and Chest Bone as mentioned in the main post are currently connected via parents. By removing the parent-child relationship from the Neck bone of the Chest bone, the problem of the Neck Bone sharing the weights goes away. 

Again this is not really an absolute solution but it works for now. I can make this a workaround by parenting the two bones via, "Offset" however making them parented via, "Connected" causes the issues of the weights.

If anyone still knows anything about this a solution is very appreciated. For now this is just a soft-fix. Thanks for all the support!